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A rigidity result for axisymmetric toric Ricci solitons

Shiqiao Zhang

Abstract

We examine a non-axisymmetric perturbation of a family of axisymmetric toric Einstein manifolds and Ricci solitons studied in Firester-Tsiamis (2024). We establish a rigidity result stating that these axisymmetric Ricci solitons do not admit constant-angle non-axisymmetric perturbations except for conformally flat cases. For these new cases, our result leads to an explicit description of the Einstein metrics and a classification of the Ricci solitons under a volume-collapsing ansatz.

A rigidity result for axisymmetric toric Ricci solitons

Abstract

We examine a non-axisymmetric perturbation of a family of axisymmetric toric Einstein manifolds and Ricci solitons studied in Firester-Tsiamis (2024). We establish a rigidity result stating that these axisymmetric Ricci solitons do not admit constant-angle non-axisymmetric perturbations except for conformally flat cases. For these new cases, our result leads to an explicit description of the Einstein metrics and a classification of the Ricci solitons under a volume-collapsing ansatz.

Paper Structure

This paper contains 5 sections, 7 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The axisymmetric case
  3. The non-axisymmetric case
  4. Rigidity result
  5. Corollaries

Key Result

Theorem 1.1

A non-axisymmetric metric eq:g is a Ricci soliton if and only if $A$ and $B$ are constants and there are functions $S^x(x, y)$ and $S^y(x, y)$ satisfying the system of equations

Theorems & Definitions (12)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 2.1: Firester--Tsiamis
  • Theorem 2.2: Firester--Tsiamis
  • Lemma 2.3: Firester--Tsiamis
  • proof : Proof of \ref{['lem:f.t.2.4']}
  • Lemma 3.1
  • proof
  • proof : Proof of \ref{['thm:main']}
  • ...and 2 more